We use on-demand AoI [1]} to measure the freshness of information seen by the users in our request-based status updating system. Let \(\Delta _k(t)\)  be the AoI [2]} about the physical quantity \(f_k\)  at the edge node at the beginning of slot \(t\) , i.e., the number of slots elapsed since the generation of the most recently received status update from sensor \(k\) .
Thus, the AoI about \(f_k\)  is a random process
\(\Delta _k(t) \triangleq t - u_k(t)\) .
We make a common assumption (e.g., [3]}, [4]}, [5]}, [6]}, [7]}, [1]}, [9]}) that \(\Delta _k(t)\)  is upper-bounded by a sufficiently large value \(\Delta ^{\mathrm {max}}\) , i.e., \(\Delta _k(t) \in \lbrace 1, 2,\ldots ,\Delta ^{\mathrm {max}}\rbrace \) .
At each slot, the AoI about \(f_k\)  drops to one if a status update from sensor \(k\)  is received and otherwise increases by one, i.e.,
\(\Delta _k(t+1)={\left\lbrace \begin{array}{ll}1,&\text{if} ~d_k(t)=1, \\\min \lbrace \Delta _k(t)+1,\Delta ^{\mathrm {max}}\rbrace ,&\text{if}~d_k(t)=0,\end{array}\right.}\) 
